Symmetry analysis of initial-value problems

Symmetry analysis is a powerful tool that enables the user to construct exact solutions of a given differential equation in a fairly systematic way. For this reason, the Lie point symmetry groups of most well-known differential equations have been catalogued. It is widely believed that the set of symmetries of an initial-value problem (or boundary-value problem) is a subset of the set of symmetries of the differential equation. The current paper demonstrates that this is untrue; indeed, an initial-value problem may have no symmetries in common with the underlying differential equation. The paper also introduces a constructive method for obtaining symmetries of a particular class of initial-value problems.     

GLOBAL ANALYSIS TO A KIND OF COMPETITION MODEL OF E-COMMERCE SITES

In this paper, we present a competition modei of e-commerce sites with the property of "better for better". Then we qualitatively analyze the special case of two dimensions. So we prove that there does not exist limit cycle, in the first quadrant of the phase plane and all trajectories tend to some singular points.     

Modelling and analysis of dynamics of viral infection of cells and of interferon resistance

Interferons are active biomolecules, which help fight viral infections by spreading from infected to uninfected cells and activate effector molecules, which confer resistance from the virus on cells. We propose a new model of dynamics of viral infection, including endocytosis, cell death, production of interferon and development of resistance. The novel element is a specific biologically justified mechanism of interferon action, which results in dynamics different from other infection models. The model reflects conditions prevailing in liquid cultures (ideal mixing), and the absence of cells or virus influx from outside. The basic model is a nonlinear system of five ordinary differential equations. For this variant, it is possible to characterise global behaviour, using a conservation law. Analytic results are supplemented by computational studies. The second variant of the model includes age-of-infection structure of infected cells, which is described by a transport-type partial differential equation for infected cells. The conclusions are: (i) If virus mortality is included, the virus becomes eventually extinct and subpopulations of uninfected and resistant cells are established. (ii) If virus mortality is not included, the dynamics may lead to extinction of uninfected cells. (iii) Switching off the interferon defense results in a decrease of the sum total of uninfected and resistant cells. (iv) Infection-age structure of infected cells may result in stabilisation or destabilisation of the system, depending on detailed assumptions. Our work seems to constitute the first comprehensive mathematical analysis of the cell-virus-interferon system based on biologically plausible hypotheses.     

Exact solutions of the Kudryashov-Sinelshchikov equation

The Kudryashov-Sinelshchikov equation for describing the pressure waves in liquid with gas bubbles is studied. New exact solutions of this equation are found. Modification of truncated expansion method is used for obtaining exact solution of this equation.     

Competitive analysis of preemptive single-machine scheduling

We give an online algorithm for minimizing the total weighted completion time on a single machine where preemption of jobs is allowed and prove that its competitive ratio is at most 1.57.     

Bridging the gap: From cellular automata to differential equation models for pedestrian dynamics

Cellular automata (CA) and ordinary differential equation (ODE) models compete for dominance in microscopic pedestrian dynamics. There are two major differences: movement in a CA is restricted to a grid and navigation is achieved by moving directly in the desired direction. Force based ODE models operate in continuous space and navigation is computed indirectly through the acceleration vector. We present the Optimal Steps Model and the Gradient Navigation Model, which produce trajectories similar to each other. Both are grid-free and free of oscillations, leading to the conclusion that the two major differences are also the two major weaknesses of the older models.     

Comment on “New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”

In this comment we analyze the paper [Abdelhalim Ebaid, S.M. Khaled, New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity, J. Comput. Appl. Math. 235 (2011) 1984-1992]. Using the traveling wave, Ebaid and Khaled have found “new types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”. We demonstrate that the authors studied the well-known nonlinear ordinary differential equation with the well-known general solution. We illustrate that Ebaid and Khaled have looked for some exact solution for the reduction of the nonlinear Schrodinger equation taking the general solution of the same equation into account.     

具转向点常微分方程混合边值问题的一致收敛差分格式

本文得到了具转向点的二阶常微分方程混合边值问题解的导数估计,提出了Il′in型差分格式,证明了此差分格式按L~1模关于小参数ε的一阶一致收敛性。最后,给出了一个数值例子,计算结果与理论分析一致。     

饮酒驾车模型

据统计,我国每天困饮酒过量而产生的车祸竟达交通事故伤亡总人数的50%-60%,酒后驾车的问题已经引起了社会各界的高度重视.为此本文建立了酒后血液中酒精含量的房室数学模型,从而为司机饮酒驾车提供合理的忠告和建议.     

A Functional EM Algorithm for Mixing Density Estimation via Nonparametric Penalized Likelihood Maximization

When the true mixing density is known to be continuous, the maximum likelihood estimate of the mixing density does not provide a satisfying answer due to its degeneracy. Estimation of mixing densities is a well-known ill-posed indirect problem. In this article, we propose to estimate the mixing density by maximizing a penalized likelihood and call the resulting estimate the nonparametric maximum penalized likelihood estimate (NPMPLE). Using theory and methods from the calculus of variations and differential equations, a new functional EM algorithm is derived for computing the NPMPLE of the mixing density. In the algorithm, maximizers in M-steps are found by solving an ordinary differential equation with boundary conditions numerically. Simulation studies show the algorithm outperforms other existing methods such as the popular EMS algorithm. Some theoretical properties of the NPMPLE and the algorithm are also discussed. Computer code used in this article is available online.     

Null controllability and global blowup controllability of ordinary differential equations with feedback controls

This paper concerns the problem of feedback null controllability and blowup controllability with feedback controls for ordinary differential equations. First, we study the feedback null controllability on a time-varying ordinary differential system by unbounded feedback operators. Then, the global exact blowup controllability with feedback controls is derived on a time-invariant ordinary differential system. Finally, we obtain the approximate null controllability by bounded feedback operators, and get the approximate blowup controllability with feedback controls for ordinary differential equations.     

一类多步并行Runge—Kutta公式

本文构造了一类适合在多处理机系统上实现的并行Ｒｕｎｇｅ－Ｋｕｔｔａ公式，对于其中的具体公式证明了收敛性，给出它的稳定区域，数值例子表明，该公式可以有效地求解常微分方程初值问题。     

Existence theorems for a class of nonlocal differential equations

A class of nonlocal second-order ordinary differential equations of the form

y^″(x)=f(x,y(x),(y○λ)(x),y^′(x))     

Competitive analysis of a dispatch policy for a dynamic multi-period routing problem

We analyze an on-line algorithm (dispatch policy) for a dynamic multi-period routing problem. The objective is to minimize the total cost over all periods. We show that the competitive ratio of this policy for instances with customers located on the non-negative real line is .     

流行性感冒的微分方程模型及数据分析

流行性感冒作为一种传染病,至今仍然经常爆发,对人们正常的工作、生活造成严重影响.依据流感的传染方式及观测数据,通过建立常微分方程模型,可探讨流感的传播规律、影响它流行的因素、预测可能发生的流行及如何抑制其流行.     

亚健康的微分方程模型及数据分析

近年来,处于亚健康状态的人越来越多,亚健康受到社会的广泛关注.依据亚健康的特征及观测数据,通过建立常微分方程模型,可探讨影响它的因素、预测可能发生的转化及如何促进亚健康向健康的转化.     

一类求解常微分方程数值解的块方法

本文讨论了一类并行计算常微分方程初值问题的带有高阶导数的块隐式混合单步方法，这种方法可以在Ｋ台处理机上并行进行数值计算，本文对方法的一般性质及收敛性进行了讨论，得知该方法的阶数为２ｌ＋１，并且指出当ｌ＝１，２时，方法是Ａ－稳定的，最后给出了一个数值例子。